Simplify the following expression and state the condition under which the simplification is valid: $a = \dfrac{z^2 - 13z + 42}{z^2 - 7z}$
Answer: First factor the expressions in the numerator and denominator. $ \dfrac{z^2 - 13z + 42}{z^2 - 7z} = \dfrac{(z - 6)(z - 7)}{(z)(z - 7)} $ Notice that the term $(z - 7)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(z - 7)$ gives: $a = \dfrac{z - 6}{z}$ Since we divided by $(z - 7)$, $z \neq 7$. $a = \dfrac{z - 6}{z}; \space z \neq 7$